Governing Singularities of Schubert Varieties
| dc.creator | Woo, Alexander | |
| dc.creator | Yong, Alexander | |
| dc.date | 2006-03-12 | |
| dc.date | 2006-06-29 | |
| dc.date.accessioned | 2026-07-07T10:02:22Z | |
| dc.date.available | 2026-07-07T10:02:22Z | |
| dc.description | We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of *interval pattern avoidance*. For "reasonable" invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P="singular"; classical pattern avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [Woo-Yong'05]; the description of the singular locus (which was independently proved by [Billey-Warrington '03], [Cortez '03], [Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with *Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures. | |
| dc.description | 23 pages. Software available at the authors' webpages. Version 2 is the submitted version. It has a nomenclature change: "Bruhat-restricted pattern avoidance" is renamed "interval pattern avoidance"; the introduction has been reorganized | |
| dc.identifier | https://arxiv.org/abs/math/0603273 | |
| dc.identifier | http://arxiv.org/abs/math/0603273 | |
| dc.identifier | J. Algebra, Vol. 320 (2008), No. 2, p. 495--520 | |
| dc.identifier | doi:10.1016/j.jalgebra.2007.12.016 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/168952 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | Combinatorics | |
| dc.subject | 14M15, 14M05, 05E99 | |
| dc.title | Governing Singularities of Schubert Varieties | |
| dc.type | text |